


The Postal Problem

by Bitenomnom



Series: Mathematical Proof [32]
Category: Sherlock (TV)
Genre: Friendship, Gen, John is sneaky, Love, Love Letters, M/M, Mathematics, Present Tense, Sherlock almost gets himself killed (again) (again), protective!John, sherlock is a postman
Language: English
Status: Completed
Published: 2012-11-02
Updated: 2012-11-02
Packaged: 2017-11-17 14:25:22
Rating: Not Rated
Warnings: Creator Chose Not To Use Archive Warnings
Chapters: 1
Words: 2,121
Publisher: archiveofourown.org
Story URL: https://archiveofourown.org/works/552537
Author URL: https://archiveofourown.org/users/Bitenomnom/pseuds/Bitenomnom
Summary: <blockquote class="userstuff">
              <p>On Sherlock's nineteenth consecutive day of temporarily being a postman, John finally sends his letter to his girlfriend. Well, not <i>quite</i> the letter he originally intended to send.</p>
            </blockquote>





	The Postal Problem

**Author's Note:**

> Okay: I know nothing about handwriting analysis, and admittedly didn't want to spend the next several hours reading about it because I (unlike Sherlock) require sleep, so...shhh...I'm aware what Sherlock says in one part is probably wrong.
> 
> Should also note that this story has very, very little to do with the math described. But I really liked the math bit, and I couldn't get the idea out of my head, so. 
> 
> And yeah, I'm going to be a jerk and let you figure the puzzle bit out for yourself. (You'll know what I mean when you see it. Also completely unrelated to the math.) It's actually really, really easy (like, not even involving the slightest bit of math, or really thought in general). Just a pain. Hopefully I haven't made any errors. And I'm sure you can get the gist of it without translating it anyway. =p
> 
> EDIT: As of 10pm CST Nov 2 2012 the code SHOULD be correct. Thank you to everyone who has continued pointing the errors out to me. (In my defense, the last one totally wasn't my fault, I think AO3 just refused to acknowledge that I changed it. >_>)
> 
> And, just for fun, perhaps don't spoil the message for anyone else who reads it, in case they're masochists like me (and, apparently, you) and also want to have to figure it out? Is that silly? It's probably silly. Oh, I dunno.

One interesting math problem is to look at a model for categorizing letters to be mailed. Say you have some sort of standard mail slot width (perhaps, for instance, five inches) and designate mail as being either large or small (for shipping/delivery price purposes) based on whether it could fit through the mail slot. Then it only matters to you whether at least _one_ of the dimensions of the mail is smaller than this minimum size (i.e. it can be as long as you want, as long as it’s skinny enough to fit through the slot).

This is known as a “classification problem.” You are trying to classify your letter as either large or small based on various data about the letter’s size.

In this case, if we make a plot of all the possible letter sizes we might have, and which qualify as large or small, we cannot find one single straight line to separate the “large” from the “small” letters. ([Here’s a picture.](http://img22.imageshack.us/img22/9168/or1e.jpg))

The fact that we have to use more than one straight line to separate two different categories means that they are not independent of one another. This makes sense, because the letter only qualifies as large if _both_ sides are larger than the slot width, meaning that you have to take into account both the length and width when determining the category. This also means that there is very strong interaction (see [Parallel](http://archiveofourown.org/works/528616)) among the attributes when determining the category. Since there’s interaction, we want to be sure to use a non-additive set function to describe what’s going on (see [Fuzzy Measures](http://archiveofourown.org/works/511646)). In this case, this means that the measure could be 0 for either of the cases where only one dimension (or neither of the dimensions) is large. The measure is only 1 when both dimensions are large. This is non-additive since 0 + 0 (the measure for a letter with large width plus the measure for a letter with large length) does not equal 1 (the measure for a letter with large width and length).

Interestingly, it is possible to draw a line through the “contours” of this plot to see it as a non-additive measure. Contours would occur if you looked at where the lines would have to be drawn for various mail slot widths. The fact that there is a corner indicates that the measure is nonadditive. (Leaving out an explanation for this bit since it would be sort of long.) ([Here’s another picture.](http://img849.imageshack.us/img849/7152/or2w.jpg))

In classification problems, what we hope to do is find the best lines to separate categories of data. We can try to find lines that will minimize the number of misclassified points (i.e. they don’t end up on the wrong side of the line), or we can try to find lines that reduce the error of the misclassified points (i.e. they’re on the wrong side, but very very close to being on the other side). Since we can’t use partial gradient searching for this (see [The (Local) Valley of Fear](http://archiveofourown.org/works/550872)), we must instead use the genetic algorithm to find the optimal lines (see [The Genetic Algorithm](http://archiveofourown.org/works/546328)).

  
  
***  
  
            Today is Sherlock’s seventeenth consecutive day of temporarily being a postman.

            John is fairly certain that if this doesn’t end soon, he is going to go mad. In fact, he’s not sure Sherlock is even doing it for a case anymore.

            “You can deduce everything about someone by the letters they send,” Sherlock says. “And—it _is_ a dying art these days—whether they send letters at all.”

            John is bloody well aware of this, yes, because he actually _has_ a letter to send to Janice, but doesn’t trust Sherlock to not intercept it before it gets there and fail to deliver it to her. He isn’t working along Baker Street, but he _is,_ whether by coincidence or not, covering her street as a part of his route.

            “Haven’t you caught the killer yet?”

            “No, John, and you’d _know_ if you’d take a few days off of work and come along.”

            “I think if we start delivering letters together we’ll look more like we’re trying to find a bachelorette party.”

            “What?”

            “Never mind.”

            Later that night, when John turns on the telly and gives up on trying convincing Sherlock to eat, John allows curiosity to take over and asks. “What can you tell about people? From their letters, I mean?”

            Sherlock’s eyes light up, and John mutes the television, bracing himself for a lengthy and energetic explanation. “Well, you can tell a great deal about a person from their handwriting alone, to start, and with something like letters it is often possible to deduce a person’s feelings about the person to whom they are sending the letter. The contents are generally easily enough determined, at least in a general sense. People do send such _dull_ things.”

            “Mm,” John agrees, or tries to agree, or tries not to agree, or whatever. He has written a letter for Janice, sort of a frivolous little thing just for the hell of it. Getting letters is fun; getting love letters is even better. John remembers his school days fondly—physical, tangible notes long before everyone and his mum was using email and text messages. John hasn’t put anything of particular importance in this letter, just stupid little things that people say to other people. Janice is…nice.

            “If I had to guess,” Sherlock continues, curling his legs up to his chest, “you, John, print the majority of your address labels in black pen. If you were writing to a business, you would use all capital letters. If you were writing to a girlfriend, your letters would be rounder. If you were writing to me, you would instead select a blue pen, and your letters would be long and sharp.”

            “You’re just making that up.”

            “Of course I’m not. Why would I do that?”

            John grins and shakes his head. No, of course he wouldn’t—Sherlock doesn’t _need_ to make things up. Sherlock doesn’t need to _act_ brilliant; he just comes that way. John turns the volume on the telly back up, and Sherlock huffs, but watches the program along with John, and then falls asleep with him on the sofa.

 

 

            On Sherlock’s eighteenth day of temporarily being a postman, he is almost strangled and stabbed.

            Then, he arrives home, and John almost does the same to him.

            “What the _hell_ were you thinking?” John all but shouts, grabbing Sherlock ‘round his collar, just where he’d been grabbed and strangled an hour before. “If you deduced that it was the murderer’s house, _why go inside?_ ” John knows the answer, of course: it’s Sherlock. He _has_ to do stupid things like that. John kicks himself for not foreseeing it and coming along, or at least following at a distance to make sure Sherlock stayed out of trouble.

            “I had to confirm it!” Sherlock protests, but weakly, as the sight of John’s furious eyes renders him more than a little silent. John is panting out panicked breaths; maybe on Sherlock’s behalf, as Sherlock hadn’t done it nearly enough at the time. All Sherlock had had the wits to do was break free and run, phone Lestrade with the address and physical description and an idea for how to apprehend her. He returned to the flat, to John, to John clutching at the front of Sherlock’s coat and nearly having a heart attack.

            “I know you’re not going to listen to me,” John says, and Sherlock cannot think of anything but how wrong John is, “but _please_ , for the love of god, never do that again.”

            “I don’t do anything ‘for the love of god,’” is the best Sherlock can think of to say in response.

            “Then for the love of _me_ ,” John breathes, and Sherlock is reasonably certain that John doesn’t know he said it, because he simply breathes, and clutches, and clutches harder as his knees wobble.

            Sherlock’s expression softens. “I’ll try, John.” It’s more than he’s ever promised anybody, which is, perhaps, not much to begin with—but it’s something.

            “At least you’re done with it,” John says, chancing a glance up at Sherlock, who nearly flinches away at the redness of John’s eyes, and then again at his words.

            “Actually, I must persist for one more day. But don’t worry: Lestrade will be just around the corner waiting to make the arrest.”

            “Sure,” John breathes, “sure.”

 

 

            On Sherlock’s nineteenth day of temporarily being a postman, John sends out his letter to Janice, albeit a slightly modified version.

            When Sherlock approaches Janice’s house to deliver her mail, he takes a good hard look at the only letter addressed to her for the day: John’s handwriting. Long and sharp. In blue.

            Sherlock opens it.

            _Knew you’d do it, you git. Don’t get killed, all right?_

_By the way, I took off work today. Incidentally, if you let the murderer get within ten feet of you, might want to shield your eyes from the blood. Did you know the tree in her back garden has an excellent view to most of the rooms on the main floor, including the entryway?_

_Just thought you’d find that interesting._

_Be careful._

_—JW_

 

            Also on Sherlock’s nineteenth day of temporarily being a postman, incidentally his first day of ending his recent service as a postman, Sherlock and Lestrade successfully apprehend the killer without any blood being splattered. For some reason, John happens to be in the area afterward.

            “Visiting Janice?” Sherlock asks, giving John an easy out.

            “No,” John says. “We broke up this morning. Guess there’s no reason to have sent that letter after all.”  
            “I didn’t deliver it, actually,” Sherlock says, features aglow.

            “Oh, well, good job on that, then.” John shrugs and grins, and together they begin pacing down the street.

Sherlock’s eyes are stuck to John, as if he is trying to puzzle out something to say to him, some way to express whatever it is that’s warming the pit of Sherlock’s stomach hot enough that the waves of heat reach his ribcage, too.

            John, oblivious—or feigning it—continues looking forward. Sherlock supposes _one_ of them has to look at the road ahead; it may as well be John. Unfazed by Sherlock’s persistent attention, John asks offhandedly, “Still happen to have that letter on you?”

            He does. Of course he does. He’s going to put it in the drawer of his bedside table upstairs. The drawer contains all the necessities: a small emergency medical kit, a knife, a small but dense chemistry handbook, a backup mobile to which Mycroft does not know the number. Far in the back of the drawer is a case left unopened for some years: also backup, also for an emergency; for something dire—cocaine, a set of sterile needles and equipment. The one or two times Sherlock has almost reached for it, though, he has been stopped, for placed atop it is a bullet dug out of the chest of one bloody awful cabbie. Sherlock intends to place this letter there with it. “I think so.”

            “Good. By the way, and you might want to listen to this…” John pulls a paper from his pocket and reads, “Forty-eight, one sixty-seven, one twenty-nine, twenty-seven, two forty-seven, one sixty, one fifty-five, two ninety-six, six, two hundred, eighty-three, seventy-six, two twenty, eighty-nine, one forty-one, one seventy-three, ninety-four, two thirty-three, seventy-five, one sixty-two, one seventy-eight, fifty-nine, one-oh-two, one fifteen, one thirty-two, eighty-eight, one sixty-nine, two eighty-three.” John stuffs the paper back into his pocket and continues walking as if he hadn’t said a word.

 

 

            Just at the change from Sherlock’s first to his second day of ending his recent service as a postman, after an afternoon of John’s letter burning in Sherlock’s pocket while he filled Lestrade in on the details, and an evening of dinner at Angelo’s at John’s insistence that he would accept nothing but Italian for dinner; after a night of sharing the sofa with John and watching John’s favorite film, letter _still_ in Sherlock’s pocket (and Sherlock knew it, and _John_ knew it, and he kept _looking_ at Sherlock with the most _indecipherable_ stare)—this night, after the film, just as the first day of Sherlock’s ending his recent day of service as a postman turns to his second, Sherlock lays the letter out on his nightstand, stares at it for two minutes, and then grins and shouts, “Eighty-eight, one sixty-nine, one sixty-seven, six, two twenty, John!”

            There is the sound of faint scrambling; Sherlock envisions John searching for his trousers, where he’d left the wrinkled paper he’d read from before. Then, there is several seconds’ silence, and the sound of feet descending stairs.

            John appears at Sherlock’s door, beaming.


End file.
